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Calculation of molecular vibrational spectra on a quantum annealer Alexander Teplukhin, Brian K. Kendrick, and Dmitri Babikov J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.9b00402 • Publication Date (Web): 17 Jul 2019 Downloaded from pubs.acs.org on July 23, 2019
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Journal of Chemical Theory and Computation
Calculation of molecular vibrational spectra on a quantum annealer Alexander Teplukhin,† Brian K. Kendrick,∗,† and Dmitri Babikov‡ †Theoretical Division (T-1, MS B221), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA ‡Department of Chemistry, Marquette University, Milwaukee, Wisconsin 53021, USA E-mail:
[email protected] Abstract Until recently molecular energy calculations using quantum computing hardware have been limited to gate-based quantum computers. In this paper, a new methodology is presented to calculate the vibrational spectrum of a molecule on a quantum annealer. The key idea of the method is a mapping of the ground state variational problem onto an Ising or quadratic unconstrained binary optimization (QUBO) problem by expressing the expansion coefficients using spins or qubits. The algorithm is general and represents a new revolutionary approach for solving the real symmetric eigenvalue problem on a quantum annealer. The method is applied to two chemically important molecules: O2 (oxygen) and O3 (ozone). The lowest two vibrational states of these molecules are computed using both a hardware quantum annealer and a software based classical annealer. Extension of the new algorithm to higher dimensions is explicitly demonstrated for an N -dimensional harmonic oscillator (N ≤ 5). The algorithm scales exponentially with dimensionality if a direct product basis is used but will exhibit polynomial scaling for a non-direct product basis.
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Introduction
Quantum computers are seen by many as a future alternative to classical computers. Although quantum supremacy has not yet been achieved, the field is advancing quite rapidly. There are two major types of quantum computing devices avaliable today: 1 quantum annealer 2 and universal quantum computer based on quantum gates. 3–5 The first type is an example of adiabatic quantum computing 6 and is used to solve optimization problems, which at first glance appears to be quite restrictive. The second type is based on quantum gates which appears to have a wider applicability and therefore may be able to simulate a larger variety of problems. However, adiabatic and gate-based quantum computing were proven to be formally equivalent. 7 Thus, the practical application space is most likely limited by the hardware realization and not necessarily by the type of approach. In either approach, the current generation of quantum computing devices has significant noise and supports a small number of fully coupled qubits (< 100). Hence, they are often referred to collectively as NISQ (Noisy Intermediate Scale Quantum) devices and their accuracy and problem size is limited. Coming from the physical chemistry community, we asked ourselves if it would be possible to program an important fundamental problem on a quantum annealer such as the commercially available D-Wave machine. 8 Typically, people who work with such devices go in the opposite direction: knowing hardware capabilities they come up with a suitable optimization problem. As a fundamental problem we chose to calculate the vibrational ground state and possibly excited states of a molecule. This problem is very important in chem9–12 13,14 istry, for example: H+ CH+ H3 O+ , H5 O+ 5 and isotopologues, 2 and deuterated n ions,
analogues, 15,16 hydrogen clusters, 17–19 their isotopologues, 20,21 hydrogen bonded systems 22 and Lennard-Jones clusters. 23,24 The common method to study these molecular systems is a Monte-Carlo (MC) method in its various flavors: variational MC, time-dependent variational MC, diffusion MC, and path integral MC. Recently, a ground state problem in electronic structure theory was implemented on both 2
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Journal of Chemical Theory and Computation
types of quantum computing devices: a quantum annealer 25,26 and a gate-based quantum computer. 27–29 In the first study, the mapping of electronic Hamiltonian to quantum annealer Hamiltonian is achieved by means of creation and annihilation fermionic operators followed by transformation to spin operators and reduction to the form that includes pairwise interactions between qubits. An iterative algorithm is used to find the lowest ground state energy. In the second work, to approach the same problem on a gate-based quantum computer, an expectation value of each term in the electronic Hamiltonian is evaluated on a trial wave function using a quantum device and the resultant total energy serves as a guide for generating the next trial wave function. The optimization step of this Variational Quantum Eigensolver (VQE), namely the trial generation, is performed on a classical computer. The iterative nature of these algorithms makes them both hybrid. In contrast to the electronic structure algorithms discussed above, the new algorithm presented in this work is general and solves any real symmetric eigenvalue problem. To our knowledge, this is the first general quantum annealer based eigenvalue solver and will be referred to below as the Quantum Annealer Eigensolver (QAE). As discussed in more detail below, our QAE algorithm is also hybrid since the variational eigenvalue problem is solved via a sequence of many quantum annealer optimizations performed with varying weights on the constraint equations (i.e., Lagrange multipliers). The scanning and optimization of the weights is done on a classical computer. Mapping the eigenvalue problem to a quantum annealer hardware is non-trivial, because the annealer solves a minimization problem defined by an Ising functional of the form H(s) =
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